SLEWING APPARATUS

Description

Technical Field

[0001] The present invention relates to a slewing apparatus that slews with a suspended load suspended from the tip of a boom.

Background Art

[0002] In a slewing apparatus that slews with a suspended load suspended from the tip of a boom, a technology of suppressing swing of the suspended load after the slewing has been known. For example, Patent Literature (hereinafter, abbreviated as PTL) 1 discloses that swing of a suspended load is suppressed by setting an acceleration interval and a deceleration interval of slewing to a time that is an integral multiple of the swing cycle of a suspended load that is in a pendulum motion. PTL 2 discloses that swing of a suspended load is suppressed by allowing each of an acceleration interval and a deceleration interval to include a constant velocity interval. US 5 823 369 A describes a crane which has a base, a slewing body, a boom, a hook and slewing actuator. The crane has a control means for breaking and stopping the slewing body. US 5 823 369 A discloses a slewing apparatus according to the preamble of the independent claim 1.

Citation List

Patent Literature

[0003] 

PTL 1
Japanese Patent No. 2501995

PTL 2
Japanese Examined Patent Application Publication No. 7-12906

Summary of Invention

Technical Problem

[0004] However, in the technologies of PTLs 1 and 2, it is necessary to set an acceleration interval and a deceleration interval equal to or longer than the swing cycle of a suspended load. This brings a problem that it is difficult to shorten the slewing time from the slewing start position to the slewing end position.

[0005] The present invention has been made in view of the aforementioned situation. An object of the present invention is to provide a slewing apparatus capable of reducing the slewing time while suppressing swing of a suspended load at the slewing end position.

Solution to Problem

[0006] According to a first aspect, the invention provides a slewing apparatus in accordance with independent claim 1. Further aspects of the invention are set forth in the dependent claims, the drawings and the following description.

(1) A slewing apparatus according the present invention includes: a base; a slewing body slewably supported by the base; a boom supported by a slewing base in a derricking and telescopic manner; a hook suspended by a rope from a tip portion of the boom; a slewing actuator that allows the slewing body to slew; and a control section that controls the slewing actuator, in which the control section performs: an acquisition process of acquiring a slewing start position and a slewing end position of the slewing body, and a pendulum length that is a length from the tip portion of the boom to a suspended load suspended from the hook; a slewing angular velocity pattern determination process of determining a slewing angular velocity pattern by optimum control in a first interval and a second interval, the slewing angular velocity pattern indicating transition of an angular velocity of the tip potion of the boom when the slewing body slews from the slewing start position to the slewing end position, the first interval being an interval in which the angular velocity is accelerated from the slewing start position, and decelerated, and accelerated to be a slewing angular velocity ω, the second interval being an interval in which the angular velocity is decelerated from the slewing angular velocity ω, and accelerated, and decelerated to be stopped at the slewing end position; and an actuator control process of controlling the slewing actuator to allow the slewing body to slew from the slewing start position to the slewing end position such that the tip portion of the boom moves in a slewing direction at a velocity indicated by the slewing angular velocity pattern, and in the slewing angular velocity pattern determination process, the slewing angular velocity pattern is determined such that in the first interval and the second interval of a control time T that is shorter than a cycle determined by the pendulum length of the suspended load that is in a pendulum motion, a difference between a maximum angular velocity and a minimum angular velocity increases as the control time T decreases.
With the configuration described above, swing in the slewing direction of the suspended load at the slewing end position can be suppressed. Further, first interval and the second interval can be shorter than a cycle To of the suspended load that is in a pendulum motion. Consequently, the slewing time from the slewing start position to the slewing end position can be reduced compared with that in the conventional method.

(2) Preferably, in the slewing angular velocity pattern determination process, the control section determines the slewing angular velocity pattern in which the control time T is shortest within a range of response performance of the slewing actuator.
With the configuration described above, the slewing time can be further reduced within the range of the response performance of the slewing actuator.

(3) For example, in the slewing angular velocity pattern determination process, the control section determines an angular velocity x'(t) of the tip portion of the boom after t second from start of slewing by specifying a coefficient a_i (i=1, ..., 5) of equation 7 satisfying an initial condition and a terminal condition of the first interval.

(4) Preferably, the slewing apparatus further includes: a derricking actuator that derricks the boom under control of the control section; and a telescopic actuator that telescopes the boom under control of the control section; in which the control section further performs a radial velocity pattern determination process of determining a radial velocity pattern, the radial velocity pattern indicating transition of a moving velocity of the tip end portion of the boom in a slewing radial direction when the slewing body slews from the slewing start position to the slewing end position, wherein in the radial velocity pattern, the slewing radius is increased and decreased in the first interval and the second interval, in the acquisition process, the control section further acquires a slewing radius r, the slewing radius r being a horizontal distance between center of slewing of the slewing body and the tip portion of the boom at the slewing start position, in the radial velocity pattern determination process, the control section determines the radial velocity pattern in which forces in the slewing radial direction acted on the suspended load at a position of the slewing radius r at end of the first interval and at end of the second interval are balanced, and in the actuator control process, the control section controls the derricking actuator and/or telescopic actuator to allow the boom to be derricked and/or telescoped such that the tip portion of the boom moves in the slewing radial direction at a velocity indicated by the radial velocity pattern.
With the configuration described above, swing in the slewing radial direction of the suspended load at the slewing end position can be suppressed.

(5) For example, in the radial velocity pattern determination process, the control section determines the radial velocity pattern in which the suspended load moves on the slewing radius r when the slewing body slews from the slewing start position to the slewing end position.

(6) As an example, in the radial velocity pattern determination process, the control section determines a moving velocity R₀'(t) in the slewing radial direction of the tip portion of the boom after t second from start of slewing by specifying a coefficient r_i (i = 0, ..., 5) of equation 12 satisfying an initial condition and a terminal condition of the first interval.

(7) As another example, in the radial velocity pattern determination process, the control section determines a moving velocity R₀'(t) in the slewing radial direction of the tip portion of the boom after t second from start of slewing by specifying a coefficient b_i (i = 1, ..., 5) of equation 19 satisfying an initial condition and a terminal condition of the first interval.

Advantageous Effects of Invention

[0007] According to the present invention, it is possible to suppress swing in the slewing direction of the suspended load at the slewing end position, and to reduce the slewing time from the slewing start position to the slewing end position.

Brief Description of Drawings

[0008] 

FIG. 1 is a schematic diagram of a rough terrain crane 10 according to the present embodiment;

FIG. 2 is a functional block diagram of the rough terrain crane 10;

FIG. 3 is a flowchart of a slewing control process;

FIG. 4 is a schematic plan view of the rough terrain crane 10;

FIG. 5A illustrates exemplary transition of the slewing angle of the boom tip portion, FIG. 5B illustrates exemplary transition of the slewing angular velocity of the boom tip portion;

FIG. 6 illustrates a crane model for determining a slewing angular velocity pattern;

FIG. 7A illustrates exemplary transition of a radial position of the boom tip portion, and FIG. 7B illustrates exemplary transition of a radial velocity of the boom tip portion;

FIG. 8 illustrates a crane model for determining a radial velocity pattern;

FIG. 9 illustrates a positional relationship between the boom tip portion and suspended load 40 in a slewing control process;

FIGS. 10A and 10B illustrate movement of suspended load 40 in the slewing control process, in which FIG. 10A illustrates the swing angle and the swing velocity in the slewing radial direction, and FIG. 10B illustrates the swing angle and the swing velocity in the slewing direction;

FIG. 11 illustrates a relationship between a coefficient α by which a cycle To for calculating a control time T is multiplied, and a slewing angular velocity pattern in a first interval; and

FIG. 12 illustrates a crane model for determining a radial velocity pattern.

Description of Embodiments

[0009] Hereinafter, a preferred embodiment of the present invention will be described with reference to the accompanying drawings as appropriate. It should be noted that the present embodiment is merely an aspect of the present invention, and it is needless to say that the embodiment may be changed without changing the scope of the present invention.

[Rough terrain crane 10]

[0010] As illustrated in FIG. 1, rough terrain crane 10 according to the present embodiment mainly includes lower traveling body 20 and upper working body 30. Lower traveling body 20 is able to travel to a destination on tires that are rotated by the driving force of an engine (not illustrated) transmitted thereto. Upper working body 30 is slewably supported by lower traveling body 20 via a slewing bearing (not illustrated). Upper working body 30 is allowed to slew relative to lower traveling body 20 by slewing motor 31 (see FIG. 2). Lower traveling body 20 is an example of a base. Upper working body 30 is an example of a slewing body. Slewing motor 31 is an example of a slewing actuator.

[0011] Upper working body 30 mainly includes telescopic boom 32, hook 33, and cabin 34. Telescopic boom 32 is derricked by derricking cylinder 35, and is telescoped by telescopic cylinder 36(see FIG. 2). Hook 33 is suspended by rope 38 extending downward from the tip portion of telescopic boom 32 (hereinafter referred to as "boom tip portion"). Hook 33 is lifted when rope 38 is wound up by winch 39 (see FIG. 2), and is lowered when rope 38 is delivered. Cabin 34 has operation section 56 (see FIG. 2) for operating lower traveling body 20 and upper working body 30.

[0012] Derricking cylinder 35 is an example of a derricking actuator. Telescopic cylinder 36 is an example of a telescopic actuator. Upper working body 30 capable of slewing relative to lower traveling body 20, slewing motor 31 allowing upper working body 30 to slew, or a slewing reduction gear not illustrated is an example of a slewing apparatus. A specific example of a slewing apparatus is not limited to rough terrain crane 10, and may be an all terrain crane, a cargo crane, or the like. Further, a base is not necessarily movable. In that case, a slewing apparatus may be a tower crane, a slewing overhead crane, or the like.

[0013] As illustrated in FIG. 2, rough terrain crane 10 includes control section 50. Control section 50 controls operation of rough terrain crane 10. Control section 50 may be implemented by a CPU (Central Processing Unit) that executes a program stored in a memory, or may be implemented by a hardware circuit, or by a combination thereof.

[0014] As illustrated in FIG. 2, control section 50 acquires various signals output from slewing angle sensor 51, derricking angle sensor 52, boom length sensor 53, rope length sensor 54, suspended load weight sensor 55, and operation section 56. Based on the acquired various signals, control section 50 controls slewing motor 31, derricking cylinder 35, telescopic cylinder 36, and winch 39.

[0015] Slewing angle sensor 51 outputs a detection signal corresponding to the slewing angle of upper working body 30 (for example, a clockwise angle where the advancing direction of lower traveling body 20 is set to be 0°). Derricking angle sensor 52 outputs a detection signal corresponding to the derricking angle of telescopic boom 32 (an angle defined by the horizontal direction and telescopic boom 32). Boom length sensor 53 outputs a detection signal corresponding to the length of telescopic boom 32 (hereinafter referred to as a "boom length"). Rope length sensor 54 outputs a detection signal corresponding to the length of the rope delivered from winch 39 (hereinafter referred to as a "delivered length"). Suspended load weight sensor 55 outputs a detection signal corresponding to the weight m (hereinafter referred to as "suspended weight m") of suspended load 40 suspended from hook 33. Strictly, the suspended weight m includes the weight of hook 33 and rope 38 extending from the boom tip portion.

[0016] Operation section 56 receives operation by a user for operating rough terrain crane 10. Then, operation section 56 outputs an operation signal corresponding to the received user operation. Specifically, control section 50 allows lower traveling body 20 to travel and allows upper working body 30 to operate based on the user operation received via operation section 56. Operation section 56 includes a lever, a steering, a pedal, an operation panel, and the like for operating rough terrain crane 10.

[0017] Operation section 56 of the present embodiment is also able to receive user operation to input a slewing end position of upper working body 30, a slewing angular velocity ω, and the like. Then, in the slewing control process described below, control section 50 allows upper working body 30 to slew, and allows telescopic boom 32 to be derricked and/or telescoped, according to a velocity pattern determined based on the input slewing end position, the slewing angular velocity ω, and the like.

[0018] Slewing motor 31, derricking cylinder 35, telescopic cylinder 36, and winch 39 of the present embodiment are hydraulic actuators. Specifically, control section 50 controls the direction and the flow rate of the hydraulic oil to be fed to thereby drive the respective actuators. However, the actuators of the present invention are not limited to hydraulic ones. They may be electric ones.

[Slewing Control Process]

[0019] Next, a slewing control process of the present embodiment will be described with reference to FIGS. 3 to 10B. The slewing control process is a process of slewing upper working body 30 from a slewing start position to a slewing end position according to a velocity pattern in which swing of suspended load 40 suspended from hook 33 at the slewing end position decreases. The slewing control process is performed by control section 50, for example.

[Acquisition Process]

[0020] First, control section 50 acquires the slewing start position, the slewing end position, the slewing angular velocity ω of upper working body 30, the derricking angle of telescopic boom 32, the boom length, the delivered length, and the suspended weight m, illustrated in FIGS. 1 to 4, via various sensors 51 to 55 and operation section 56 (S11). The process of step S11 is an example of an acquisition process.

[0021] The slewing start position is a current position of upper working body 30, for example. Specifically, control section 50 may acquire the slewing start position based on a detection signal output from slewing angle sensor 51. The slewing end position is a position of upper working body 30 after the slewing control process ends. The slewing angular velocity ω indicates a slewing angular velocity of upper working body 30 in a constant velocity interval described below. Control section 50 may acquire the slewing end position and the slewing angular velocity ω from the user via operation section 56. However, if an input of the slewing angular velocity ω is omitted, a preset default slewing angular velocity ω may be used.

[0022] Further, control section 50 calculates a slewing radius r at the slewing start position based on the derricking angle and the boom length. The slewing radius r indicates a horizontal distance between the slewing center of upper working body 30 and the boom tip portion, for example. The boom tip portion is a position of the center of rotation of a sheave for winding rope 38, for example. Control section 50 also calculates a pendulum length 1 that is a length from the boom tip portion to suspended load 40, based on the boom length and the delivered length. Control section 50 may calculate the pendulum length 1 by adding a predetermined constant corresponding to the length from hook 33 to the position of the center of gravity of suspended load 40, to the length between the boom tip portion and hook 33 calculated based on the boom length and the delivered length, for example.

[Slewing angular velocity pattern determination process]

[0023] Next, control section 50 determines a slewing angular velocity pattern (S12). The slewing angular velocity pattern represents transition of the angular velocity of the boom tip portion when upper working body 30 slews. As illustrated in FIG. 5B, the slewing angular velocity pattern includes a first interval of a control time T from the slewing start position until it reaches the slewing angular velocity ω, a constant velocity interval in which moving is performed constantly at the slewing angular velocity ω, and a second interval of the control time T from the slewing angular velocity ω until it stops at the slewing end position. The process of step S12 is an example of a slewing angular velocity pattern determination process.

[0024] In more detail, the boom tip portion is accelerated from a velocity 0 in the first interval of the control time, then decelerated, and then accelerated to reach the slewing angular velocity ω. In the below description, an angular velocity at the time of switching the velocity from acceleration to deceleration is referred to as a "maximum angular velocity", and an angular velocity at the time of switching the velocity from deceleration to acceleration is referred to as a "minimum angular velocity". In the example of FIG. 5B, the maximum angular velocity is ω, and the minimum angular velocity is 0. Then, in the slewing angular velocity pattern in the first interval, a difference between the maximum angular velocity and the minimum angular velocity increases as the control time T is shorter. In other words, the boom tip portion in the first interval is rapidly accelerated, rapidly decelerated, and rapidly accelerated, as the control time T is shorter.

[0025] The control time T is determined as described below, for example. First, control section 50 considers rope 38, hook 33, and suspended load 40, extending from the boom tip portion, as a pendulum, and calculates a cycle T₀ of the pendulum according to equation 1. Next, control section 50 calculates the control time T (= T₀ × α) by multiplying the cycle T₀ by a coefficient α (α < 1). The coefficient α is a value determined according to the response performance of slewing motor 31, for example. Specifically, the coefficient α may be decreased within a range that slewing motor 31 can follow the slewing angular velocity pattern when the coefficient α is decreased (that is, the control time T is shortened). In the present embodiment, the coefficient α = 0.4.
[1]

[0026] Further, the slewing angular velocity pattern in the second interval is in rotational symmetry with the slewing angular velocity pattern of the first interval, for example. Specifically, in the second interval of the control time T, the boom tip portion is decelerated from the slewing angular velocity ω, then accelerated, then decelerated, and then stops at the slewing end position. Hereinafter, a procedure of determining the slewing angular velocity pattern of the first interval will be described in detail.

[0027] First, control section 50 analytically derives a moving locus of the boom tip portion in the slewing direction with use of a crane model illustrated in FIG. 6. In FIG. 6, x represents a position of the boom tip portion moving from the initial position O (that is, position of the boom tip portion corresponding to the slewing start position). θ represents an angle (hereinafter referred to as a "pendulum angle") between rope 38 extending from the boom tip portion at the position x and the vertical direction, g represents gravitational acceleration. An equation of motion of the crane model illustrated in FIG. 6 is expressed by equation 2 provided below. Further, equation 3 is established by linearize equation 2.
[2]


[3]

[0028] Next, with use of equation 3 as a control target, a locus of the boom tip portion in the slewing direction is designed by using an evaluation function of the optimum control theory expressed by equation 4. Specifically, by extending equation 4 by Lagrange multiplier method so as to include equation 3 as a constrain condition, equation 5 is established. Further, an integrand F₁', when a functional J₁ is minimized, satisfies equation 6. Then, by solving it, equation 7 is obtained.
[4]


[5]


[6]


[7]

[0029] Here, λ₁ of equation 5 represents an undefined multiplier of Lagrange. Further, a constant a_i (i = 1, ..., 5) of equation 7 is specified when being applied with the initial condition and the terminal condition expressed in equation 8. Specifically, when equation 6 in which z₁ is substituted with x is solved for x, x', equation 6 in which z₁ is substituted with θ is solved for θ, θ', and equation 6 in which z₁ is substituted with λ₁ is solved for λ₁, five equations including undefined constants a₁ to as obtained in the process of integration are obtained. By assigning the respective conditions of equation 8 to the obtained five equations to solve the simultaneous equations, the constants a₁ to as are specified. For example, in the slewing angular velocity pattern illustrated in FIG. 5B, a₁ = 0.6609, a₂ = 2.034, a₃ = 0, a₄ = 1.743, and as = -20.53, for example. Further, R₀(T) represents a slewing radius after T seconds from the start of slewing, which is calculated from equation 9.
[8]

[Radial velocity pattern determination process]

[0030] Next, control section 50 determines a radial velocity pattern (S13). The radial velocity pattern shows transition of the moving velocity in the slewing radius direction of the boom tip portion when upper working body 30 slews from the slewing start position to the slewing end position. According to an example of a radial velocity pattern illustrated in FIG. 7B, the boom tip portion in the first interval is moved in a direction of increasing the slewing radius, and then, moved in a direction of decreasing the slewing radius. Meanwhile, the boom tip portion in the constant velocity interval is not moved in the slewing radius direction. The radial velocity pattern in the second interval is in rotational symmetric to the radial velocity pattern in the first interval. The process of step S13 is an example of a radial velocity pattern determination process.

[0031] In more detail, the boom tip portion in the first interval is moved from the position of the slewing radius r at the moving start position in a direction of increasing the slewing radius, and then, moved in a direction of decreasing the slewing radius, and then reaches the position of a target slewing radius r', described below, at the end of the first interval. The radial velocity pattern in the first interval defines a moving pattern of the boom tip portion for balancing forces in the slewing radius direction (that is, centrifugal force and a horizontal component of the tensile force of rope 38) acted on suspended load 40 at the position of the slewing radius r, at the end of the first interval.

[0032] Meanwhile, the boom tip portion in the constant velocity interval is not moved in the slewing radius direction from the target slewing radius r'. Specifically, the magnitude of the horizontal component of the tensile force of rope 38 acted on suspended load 40 is not changed in the constant velocity interval. As the slewing angular velocity ω of suspended load 40 in the constant velocity interval is constant, the centrifugal force acted on suspended load 40 is not changed either. Consequently, suspended load 40 in the constant velocity interval moves on the position of the slewing radius r in a state where the forces in the slewing radial direction are balanced, as illustrated in FIG. 9 by a solid line.

[0033] Further, the boom tip portion in the second interval is moved from the position of the target slewing radius r' up to a position where the slewing radius is greater than that at the position of the slewing radius r, and then, moved in a direction of decreasing the slewing radius, and reaches the position of the slewing radius r at the end of the second interval (that is, moving end position). The radial velocity pattern in the second interval defines a moving pattern of the boom tip portion for causing the forces in the slewing radial direction (that is, centrifugal force, and the horizontal component of the tensile force of rope 38) to be zero in suspended load 40 at the position of the slewing radius r, at the end of the second interval,.

[0034] The target slewing radius r' is determined as described below, for example. In the crane model of FIG. 8, the target slewing radius r' for balancing the forces in the slewing radial direction acted on suspended load 40 at the position of the slewing radius r is calculated by equation 9, for example. Further, ϕe in equation 9 represents a pendulum angle at the end of the first interval, which is calculated by equation 10.
[9]


[10]

[0035] Then, control section 50 sets R₀(t), representing transition of the slewing radius in the first interval, as a fifth function as expressed by equation 11. Then, by differentiating R₀(t), a radial velocity pattern expressed by equation 12 is obtained.
[11]


[12]

[0036] It should be noted that the constant r_i (i = 0, ..., 5) in equations 11 and 12 is specified by applying the initial condition, the boundary condition, and the terminal condition of equation 13. Specifically, it is only necessary to solve simultaneous equations by applying the respective conditions of equation 13 to equations 11 and 12. For example, in the radial velocity pattern illustrated in FIG. 7B, r₀ = 10.08, r₁ = 0, r₂ = 1.355, r₃ = -1.770, r₄ = 0.6424, and r₅ = -0.07070.
[13]

[Actuator control process]

[0037] Next, control section 50 drives slewing motor 31 according to the determined slewing angular velocity pattern. Control section 50 also drives derricking cylinder 35 and/or the telescopic cylinder 36 according to the determined radial velocity pattern (S14). The process of step S14 is an example of an actuator control process.

[0038] Specifically, control section 50 controls slewing motor 31 to allow upper working body 30 to slew from the slewing start position to the slewing end position such that the boom tip portion moves in the slewing direction at an angular velocity indicated by a slewing angular velocity pattern. FIG. 5A illustrates transition of the slewing angle at the boom tip portion that moves according to the slewing angular velocity pattern illustrated in FIG. 5B.

[0039] Control section 50 also controls derricking cylinder 35 and/or telescopic cylinder 36 to allow telescopic boom 32 to be derricked and/or telescoped such that the boom tip portion moves in the slewing radial direction at a velocity shown by the radial velocity pattern. FIG. 7A illustrates transition of the position in the slewing radial direction of the boom tip portion that moves according to the radial velocity pattern illustrated in FIG. 7B.

[0040] It should be noted that control section 50 may realize movement of the boom tip portion according to the radial velocity pattern by one of derricking cylinder 35 and telescopic cylinder 36, or may be realized by both derricking cylinder 35 and telescopic cylinder 36. For example, control section 50 may select an actuator to be used for realizing the radial velocity pattern according to the derricking angle of telescopic boom 32 at the slewing start position.

[0041] When the derricking angle of telescopic boom 32 is smaller than a first threshold, control section 50 may control operation in the slewing radial direction by only using telescopic cylinder 36. Further, when the derricking angle of telescopic boom 32 is equal to or larger than the first threshold but smaller than a second threshold, control section 50 may control operation in the slewing radial direction by linking derricking cylinder 35 and telescopic cylinder 36. Furthermore, when the derricking angle of telescopic boom 32 is equal to or larger than the second threshold, control section 50 may control operation in the slewing radial direction by only using derricking cylinder 35. It should be noted that the second threshold is larger than the first threshold. For example, it is acceptable that first threshold = 30° and second threshold = 60°.

[0042] When attempting to realize the radial velocity pattern by using both derricking cylinder 35 and telescopic cylinder 36, control section 50 may resolve the radial velocity pattern into a derricking velocity and a telescopic velocity. Then, control section 50 may drive derricking cylinder 35 according to the derricking velocity, and drive telescopic cylinder 36 according to the telescopic velocity.

[Action and effect of embodiment]

[0043] FIG. 9 illustrates a positional relationship in the slewing radial direction between the boom tip portion and suspended load 40 when the boom tip portion is moved according to the slewing angular velocity pattern illustrated in FIG. 5B and the radial velocity pattern illustrated in FIG. 7B. Suspended load 40 illustrated by a solid line in FIG. 9 moves on the circumference of the slewing radius r. Meanwhile, the position of the boom tip portion illustrated by a dotted line in FIG. 9 moves on the circumference of the target slewing radius r' that is smaller than the slewing radius r in the constant velocity interval. Then, the position of the boom tip portion in the slewing radial direction overlaps the position of suspended load 40 in the slewing radial direction at the start of the first interval and the end of the second interval.

[0044] FIG. 10A illustrates a relationship between the swing angle (solid line) of suspended load 40 in the slewing radial direction and the swing velocity (dotted line) of suspended load 40 in the slewing radial direction, and FIG. 10B illustrates a relationship between the swing angle (solid line) of suspended load 40 in the slewing direction and the swing velocity (dotted line) of suspended load 40 in the slewing direction, when the boom tip portion is moved according to the slewing angular velocity pattern illustrated in FIG. 5B and the radial velocity pattern illustrated in FIG. 7B. It should be noted that the swing angle indicates an angle defined by the vertical direction and rope 38. Further, the swing velocity indicates a relative velocity (velocity difference) to the velocity of the boom tip portion.

[0045] As illustrated in FIG. 10A, suspended load 40 in the first interval and the second interval swings in the slewing radial direction when the boom tip portion is moved in the slewing radial direction according to the radial velocity pattern. Then, at the end of the first interval, the swing velocity of suspended load 40 in the slewing radial direction converges to almost zero, and the swing angle of suspended load 40 in the slewing radial direction converges to almost ϕe. In the constant velocity interval, the swing velocity of suspended load 40 in the slewing radial direction is stable at almost zero, and the swing angle of suspended load 40 in the slewing radial direction is stable at almost ϕe. Then, at the end of the second interval, the swing velocity of suspended load 40 in the slewing radial direction converges to almost zero, and the swing angle of suspended load 40 in the slewing radial direction converges to almost zero.

[0046] As illustrated in FIG. 10B, in the first interval and the second interval, suspended load 40 swings in the slewing direction when the boom tip portion is moved in the slewing direction according to the slewing angular velocity pattern. Then, at the end of the first interval and at the end of the second interval, the swing velocity of suspended load 40 in the slewing direction converges to almost zero, and the swing angle of suspended load 40 in the slewing direction converges to almost zero. Further, in the constant velocity interval, the swing velocity of suspended load 40 in the slewing direction is stable at almost zero, and the swing angle of suspended load 40 in the slewing direction is stable at almost zero.

[0047] As described above, according to the aforementioned embodiment, it is possible to suppress not only the swing of suspended load 40 in the slewing direction at the slewing end position but also the swing of suspended load 40 in the slewing radial direction. Consequently, when telescopic boom 32 is allowed to slew in a narrow space in particular, it is possible to suppress suspended load 40, pushed out by the centrifugal force, from being brought into contact with an obstacle.

[0048] Further, according to the aforementioned embodiment, the control time T of the first interval and the second interval can be reduced from the cycle T₀ of suspended load 40 performing pendulum motion within a range of response performance of slewing motor 31. Consequently, the slewing time from the slewing start position to the slewing end position can be reduced. It should be noted that in the slewing angular velocity pattern, the constant velocity interval is not indispensable, and may be omitted.

[0049] FIG. 11 illustrates a relationship between the coefficient α for calculating the control time T and the slewing angular velocity pattern in the first interval. In FIG. 11, the slewing angular velocity pattern where α = 0.4 (T = 0.4 T₀) is illustrated by a solid line, the slewing angular velocity pattern where α = 0.6 (T = 0.6 T₀) is illustrated by a broken line, the slewing angular velocity pattern where α = 0.8 (T = 0.8 T₀) is illustrated by alternate long and short dashed lines, and the slewing angular velocity pattern where α = 1 (T = T₀) is illustrated by alternate long and two short dashed lines.

[0050] As illustrated in FIG. 11, as the coefficient α is smaller, the control time T taken until the angular velocity ω is reached is shorter. Accordingly, from the viewpoint of reducing the slewing time from the slewing start position to the slewing end position, a smaller coefficient α value is desirable. On the other hand, as the value of coefficient α is smaller, a difference between the maximum angular velocity and the minimum angular velocity increases, whereby sudden acceleration and sudden deceleration are required. In other words, as the value of coefficient α is larger, a difference between the maximum angular velocity and the minimum angular velocity decreases, whereby the slewing angular velocity pattern where coefficient α = 1 becomes a straight line (that is, uniformly accelerated motion).

[0051] Specifically, when the value of coefficient α is too small, even if control section 50 attempts to control slewing motor 41 according to the slewing angular velocity pattern, slewing motor 41 may not follow. As such, it is desirable to select a minimum coefficient α within the range of the response performance of slewing motor 41. It should be noted that the response performance of slewing motor 41 may also include the response performance of a valve and the like disposed on the oil passage for feeing hydraulic oil to slewing motor 41, in addition to the response performance itself of slewing motor 41.

[0052] While the aforementioned embodiment has described an example in which a radial velocity pattern is determined according to equation 12, the method of determining a radial velocity pattern is not limited to this. It may be determined by optimum control like a slewing angular velocity pattern. Specifically, the equation of motion of the crane model illustrated in FIG. 12 may be expressed as equation 14. Further, equation 15 is established by approximating equation 14. It should be noted that constant Ω in equation 15 corresponds to the slewing angular velocity of the centrifugal force term of equation 14.
[14]


[15]

[0053] Then, with use of equation 15 as a control target, a locus of the boom tip portion in the slewing radial direction is designed by using an evaluation function of the optimum control theory expressed by equation 16. Specifically, by extending equation 16 by Lagrange multiplier method so as to include equation 15 as a constrain condition, equation 17 is established. Further, an integrand F2', when a functional J₂ is minimized, satisfies equation 18. Then, by solving it, equation 19 is obtained.
[16]


[17]


[18]


[19]

[0054] Here, λ₁ of equation 17 is an undefined multiplier of Lagrange. Further, a constant b_i (i = 1, ..., 5) of equation 19 is specified when being applied with the initial condition and the terminal condition expressed in equation 20. Specifically, when equation 18 in which z₂ is substituted with Ro is solved for Ro, Ro', equation 18 in which z₂ is substituted with ϕ is solved for ϕ,ϕ', and equation 18 in which z₂ is substituted with λ₂ is solved for λ₂, five equations including undefined constants b₁ to bs, obtained in the process of integration, are established. By assigning the respective conditions of equation 20 to the established five equations to solve the simultaneous equations, the constants b₁ to b₅ are specified. For example, in the radial velocity pattern illustrated in FIG. 7B, b₁ = 46.22, b₂ = -104.8, b₃ = 96.34, b₄ = -119.0, and b₅ = -50.62, for example. Further, constant Ω is a value derived by trial and error in order to obtain a preferable radial velocity pattern. For example, in the radial velocity pattern illustrated in FIG. 7B, Ω = 1.5rpm.
[20]

Reference Signs List

[0055] 

10

Rough terrain crane

20

Lower traveling body

30

Upper working body

31

Slewing motor

32

Telescopic boom

33

Rope

36

Derricking cylinder

37

Telescopic cylinder

38

Rope

50

Control section

51

Slewing angle sensor

52

Derricking angle sensor

53

Boom length sensor

54

Rope length sensor

55

Suspended load weight sensor

56

Operation section

Claims

1. A slewing apparatus comprising:

a base (20);

a slewing body (30) slewably supported by the base (20);

a boom (32) supported by a slewing base (30) in a derricking and telescopic manner;

a hook (33) suspended by a rope (38) from a tip portion of the boom (32);

a slewing actuator (31) that allows the slewing body (30) to slew; and

a control section (50) that controls the slewing actuator (31), wherein

the control section (50) performs

an acquisition process of acquiring a slewing start position and a slewing end position of the slewing body (30), and a pendulum length (1) that is a length from the tip portion of the boom (32) to a suspended load suspended from the hook (33);
characterized in that

the control section (50) performs:

a slewing angular velocity pattern determination process of determining a slewing angular velocity pattern by optimum control in a first interval and a second interval, the slewing angular velocity pattern indicating transition of an angular velocity of the tip potion of the boom (32) when the slewing body (30) slews from the slewing start position to the slewing end position, the first interval being an interval in which the angular velocity is accelerated from the slewing start position, and decelerated, and accelerated to be a slewing angular velocity ω, the second interval being an interval in which the angular velocity is decelerated from the slewing angular velocity ω, and accelerated, and decelerated to be stopped at the slewing end position; and

an actuator control process of controlling the slewing actuator (31) to allow the slewing body (30) to slew from the slewing start position to the slewing end position such that the tip portion of the boom (32) moves in a slewing direction at a velocity indicated by the slewing angular velocity pattern, wherein

in the slewing angular velocity pattern determination process, the slewing angular velocity pattern is determined such that in the first interval and the second interval of a control time T that is shorter than a cycle determined by the pendulum length of the suspended load that is in a pendulum motion, a difference between a maximum angular velocity and a minimum angular velocity increases as the control time T decreases.

2. The slewing apparatus according to claim 1, wherein
in the slewing angular velocity pattern determination process, the control section (50) determines the slewing angular velocity pattern in which the control time T is shortest within a range of response performance of the slewing actuator (31).

3. The slewing apparatus according to claim 1 or 2, wherein
in the slewing angular velocity pattern determination process, the control section (50) determines an angular velocity x'(t) of the tip portion of the boom (32) after t second from start of slewing by specifying a coefficient a_i (i=1, ..., 5) of equation 1 satisfying an initial condition and a terminal condition of the first interval.
[1]

4. The slewing apparatus according to any one of claims 1 to 3, further comprising:

a derricking actuator (35) that derricks the boom (32) under control of the control section (50); and

a telescopic actuator (36) that telescopes the boom (32) under control of the control section (50); wherein

the control section (50) further performs a radial velocity pattern determination process of determining a radial velocity pattern, the radial velocity pattern indicating transition of a moving velocity of the tip end portion of the boom (32) in a slewing radial direction when the slewing body (30) slews from the slewing start position to the slewing end position, wherein in the radial velocity pattern, the slewing radius is increased and decreased in the first interval and the second interval,

in the acquisition process, the control section (50) further acquires a slewing radius r, the slewing radius r being a horizontal distance between center of slewing of the slewing body (30) and the tip portion of the boom (32) at the slewing start position,

in the radial velocity pattern determination process, the control section (50) determines the radial velocity pattern in which forces in the slewing radial direction acted on the suspended load at a position of the slewing radius r at end of the first interval and at end of the second interval are balanced, and

in the actuator control process, the control section (50) controls the derricking actuator (35) and/or telescopic actuator (36) to allow the boom (32) to be derricked and/or telescoped such that the tip portion of the boom (32) moves in the slewing radial direction at a velocity indicated by the radial velocity pattern.

5. The slewing apparatus according to claim 4, wherein
in the radial velocity pattern determination process, the control section (50) determines the radial velocity pattern in which the suspended load moves on the slewing radius r when the slewing body (30) slews from the slewing start position to the slewing end position.

6. The slewing apparatus according to claim 4 or 5, wherein
in the radial velocity pattern determination process, the control section (50) determines a moving velocity R₀'(t) in the slewing radial direction of the tip portion of the boom (32) after t second from start of slewing by specifying a coefficient r_i (i = 0, ..., 5) of equation 2 satisfying an initial condition and a terminal condition of the first interval.
[2]

7. The slewing apparatus according to claim 4 or 5, wherein
in the radial velocity pattern determination process, the control section (50) determines a moving velocity R₀'(t) in the slewing radial direction of the tip portion of the boom (32) after t second from start of slewing by specifying a coefficient b_i (i = 1, ..., 5) of equation 3 satisfying an initial condition and a terminal condition of the first interval.
[3]

Ansprüche

1. Schwenkvorrichtung, umfassend:

eine Basis (20);

einen Schwenkkörper (30), der schwenkbar von der Basis (20) gestützt wird;

einen Arm (32), der von einer Schwenkbasis (30) wippenartig und teleskopartig gestützt wird;

einen Haken (33), der an einem Seil (38) von einem Spitzenbereich des Arms (32) aufgehängt ist;

einen Schwenkaktuator (31), der es dem Schwenkkörper ermöglicht zu schwenken; und

einen Steuerabschnitt (50), der den Schwenkaktuator (31) steuert, wobei

der Steuerabschnitt (50) ausführt

einen Erfassungsprozess eines Erfassens einer Schwenkstartposition und einer Schwenkendposition des Schwenkkörpers (30), und einer Pendellänge (1), die eine Länge von dem Spitzenbereich des Arms (32) zu einer an dem Haken (33) aufgehängten Last ist;

dadurch gekennzeichnet, dass

der Steuerabschnitt (50) ausführt:

einen Schwenkwinkelgeschwindigkeitsmuster-Bestimmungsprozess eines Bestimmens eines Schwenkwinkelgeschwindigkeitsmusters durch eine Optimumsteuerung in einem ersten Intervall und in einem zweiten Intervall, wobei das Schwenkwinkelgeschwindigkeitsmuster einen Übergang einer Schwenkwinkelgeschwindigkeit des Spitzenbereichs des Arms (32) angibt, wenn der Schwenkkörper (30) von der Schwenkstartposition in die Schwenkendposition schwenkt, wobei das erste Intervall ein Intervall ist, in dem die Winkelgeschwindigkeit von der Schwenkstartposition beschleunigt wird, und verlangsamt, und beschleunigt, um eine Schwenkwinkelgeschwindigkeit ω zu erreichen, wobei das zweite Intervall ein Intervall ist, in dem die Winkelgeschwindigkeit ausgehend von der Schwenkwinkelgeschwindigkeit ω verlangsamt wird, und beschleunigt, und verlangsamt, um an der Schwenkwinkelendposition gestoppt zu werden; und

einen Aktuatorsteuerprozess eines Steuerns des Schwenkaktuators (31), um es dem Schwenkkörper (30) zu ermöglichen, von der Schwenkstartposition in die Schwenkendposition zu so schwenken, dass sich der Spitzenbereich des Arms (32) in eine Schwenkrichtung bei einer Geschwindigkeit bewegt, die durch das Schwenkwinkelgeschwindigkeitsmuster angegeben ist, wobei

in dem Schwenkwinkelgeschwindigkeitsmuster-Bestimmungsprozess das Schwenkwinkelgeschwindigkeitsmuster so bestimmt wird, dass sich, in dem ersten Intervall und dem zweiten Intervall einer Steuerzeit T, die kürzer ist als ein Zyklus, der durch die Pendellänge der aufgehängten Last bestimmt ist, die in einer Pendelbewegung ist, eine Differenz zwischen einer maximalen Winkelgeschwindigkeit und einer minimalen Winkelgeschwindigkeit vergrößert, wenn sich die Steuerzeit T verringert.

2. Schwenkvorrichtung nach Anspruch 1, wobei
in dem Schwenkwinkelgeschwindigkeitsmuster-Bestimmungsprozess der Steuerabschnitt (50) das Schwenkwinkelgeschwindigkeitsmuster bestimmt, in dem die Steuerzeit T innerhalb eines Bereichs einer Antwortperformanz des Schwenkaktuators (31) am kürzesten ist.

3. Schwenkvorrichtung nach Anspruch 1 oder 2, wobei
in dem Schwenkwinkelgeschwindigkeitsmuster-Bestimmungsprozess der Steuerabschnitt (50) eine Winkelgeschwindigkeit x'(t) des Spitzenbereichs des Arms (32) nach t Sekunden ab einem Start eines Schwenkens durch Spezifikation eines Koeffizienten a_i (i=1, ..., 5) aus Gleichung 1 bestimmt, der eine Anfangsbedingung und eine Endbedingung des ersten Intervalls erfüllt.
[1]

4. Schwenkvorrichtung nach einem der Ansprüche 1 bis 3, weiter umfassend:

einen Wippaktuator (35), der den Arm (32) unter der Steuerung des Steuerabschnitts (50) wippt; und

einen Teleskopaktuator, der den Arm (32) unter der Steuerung des Steuerabschnitts (50) teleskopiert; wobei

der Steuerabschnitt (50) weiter einen Radialgeschwindigkeitsmuster-Bestimmungsprozess eines Bestimmens eines Radialgeschwindigkeitsmusters ausführt, wobei das Radialgeschwindigkeitsmuster einen Übergang einer Bewegungsgeschwindigkeit des Spitzenendbereichs des Arms (32) in eine Schwenkradialrichtung angibt, wenn der Schwenkkörper (30) von der Schwenkstartposition in die Schwenkendposition schwenkt, wobei in dem Radialgeschwindigkeitsmuster der Schwenkradius in dem ersten Intervall und in dem zweiten Intervall erhöht wird und verringert wird,

in dem Erfassungsprozess der Steuerabschnitt (50) weiter einen Schwenkradius r erfasst, wobei der Schwenkradius r eine horizontale Distanz zwischen einem Zentrum eines Schwenkens des Schwenkkörpers (30) und dem Spitzenbereich des Arms (32) in der Schwenkstartposition ist,

in dem Radialgeschwindigkeitsmuster-Bestimmungsprozess der Steuerabschnitt das Radialgeschwindigkeitsmuster bestimmt wird, in dem Kräfte in der Schwenkradialrichtung, die auf die aufgehängte Last wirkten, bei einer Position des Schwenkradius r am Ende des ersten Intervalls und am Ende des zweiten Intervalls ausgeglichen sind, und

in dem Aktuatorsteuerprozess der Steuerabschnitt (50) den Wippaktuator (35) und/oder den Teleskopaktuator (36) steuert, um es dem Arm (32) zu ermöglichen, so gewippt oder teleskopiert zu werden, dass sich der Spitzenbereich des Arms (32) in die Schwenkradialrichtung bei einer Geschwindigkeit bewegt, die durch das Radialgeschwindigkeitsmuster gegeben ist.

5. Schwenkvorrichtung nach Anspruch 4, wobei
in dem Radialgeschwindigkeitsmuster-Bestimmungsprozess der Steuerabschnitt (50) das Radialgeschwindigkeitsmuster bestimmt, in dem sich die aufgehängte Last auf dem Schwenkradius r bewegt, wenn der Schwenkkörper (30) von der Schwenkstartposition in die Schwenkendposition schwenkt.

6. Schwenkvorrichtung nach Anspruch 4 oder 5, wobei
in dem Radialgeschwindigkeitsmuster-Bestimmungsprozess der Steuerabschnitt (50) eine Bewegungsgeschwindigkeit R₀'(t) in der Schwenkradialrichtung des Spitzenbereichs des Arms (32) nach t Sekunden nach dem Start des Schwenkens durch Spezifikation eines Koeffizienten r_i (i = 0, ..., 5) aus Gleichung 2 bestimmt, der eine Anfangsbedingung und eine Endbedingung des ersten Intervalls erfüllt.
[2]

7. Schwenkvorrichtung nach Anspruch 4 oder 5, wobei
in dem Radialgeschwindigkeitsmuster-Bestimmungsprozess der Steuerabschnitt (50) eine Bewegungsgeschwindigkeit R₀'(t) in der Schwenkradialrichtung des Spitzenbereichs des Arms (32) nach t Sekunden nach dem Start des Schwenken durch Spezifikation eines Koeffizienten b_i (i = 1, ..., 5) aus Gleichung 3 bestimmt, der eine Anfangsbedingung und eine Endbedingung des ersten Intervalls erfüllt.
[3]

Revendications

1. Dispositif de pivotement comprenant :

une base (20) ;

un corps pivotant (30) supporté de manière pivotante par la base (20) ;

une flèche (32) supportée par une base pivotante (30) de manière relevable et télescopique ;

un crochet (33) suspendu par une corde (38) à une partie de pointe de la flèche (32) ;

un actionneur de pivotement (31) qui permet au corps pivotant (30) de pivoter ; et

une unité de contrôle (50) qui contrôle l'actionneur de pivotement (31), dans lequel l'unité de contrôle (50) effectue

un procédé d'acquisition pour acquérir une position de début de pivotement et une position de fin de pivotement du corps pivotant (30), et une longueur de pendule (1) qui est une longueur de la partie de pointe de la flèche (32) jusqu'à une charge suspendue au crochet (33) ;

caractérisé en ce que :

l'unité de contrôle (50) effectue

un procédé de détermination de profil de vitesse angulaire de pivotement pour déterminer un profil de vitesse angulaire de pivotement par commande optimale dans un premier intervalle et un second intervalle, le profil de vitesse angulaire de pivotement indiquant la transition d'une vitesse angulaire de la partie de pointe de la flèche (32) lorsque le corps pivotant (30) pivote de la position de début de pivotement à la position de fin de pivotement, le premier intervalle étant un intervalle dans lequel la vitesse angulaire est accélérée à partir de la position de début de pivotement, et décélérée, et accélérée pour atteindre une vitesse angulaire de pivotement ω, le second intervalle étant un intervalle dans lequel la vitesse angulaire est décélérée à partir de la vitesse angulaire de pivotement ω, et accélérée, et décélérée pour être arrêtée à la position de fin de pivotement ; et

un procédé de contrôle d'actionneur pour contrôler l'actionneur de pivotement (31) afin de permettre au corps pivotant (30) de pivoter de la position de début de pivotement à la position de fin de pivotement de telle sorte que la partie de pointe de la flèche (32) se déplace dans une direction de pivotement à une vitesse indiquée par le profil de vitesse angulaire de pivotement, dans lequel

dans le procédé de détermination de profil de vitesse angulaire de pivotement, le profil de vitesse angulaire de pivotement est déterminé de telle sorte que dans le premier intervalle et le second intervalle d'un temps de contrôle T qui est plus court qu'un cycle déterminé par la longueur de pendule de la charge suspendue qui est dans un mouvement de pendule, une différence entre une vitesse angulaire maximale et une vitesse angulaire minimale augmente à mesure que le temps de contrôle T diminue.

2. Dispositif de pivotement selon la revendication 1, dans lequel
dans le procédé de détermination de profil de vitesse angulaire de pivotement, l'unité de contrôle (50) détermine le profil de vitesse angulaire de pivotement dans lequel le temps de contrôle T est le plus court à l'intérieur d'une plage de performance de réponse de l'actionneur de pivotement (31).

3. Dispositif de pivotement selon la revendication 1 ou 2, dans lequel
dans le procédé de détermination de profil de vitesse angulaire de pivotement, l'unité de contrôle (50) détermine une vitesse angulaire x'(t) de la partie de pointe de la flèche (32) après t seconde à partir du début du pivotement en spécifiant un coefficient a_i (i= 1, ..., 5) de l'équation 1 satisfaisant une condition initiale et une condition terminale du premier intervalle.
[1]

4. Dispositif de pivotement selon l'une quelconque des revendications 1 à 3, comprenant en outre :

un actionneur de relevage (35) qui relève la flèche (32) sous le contrôle de l'unité de contrôle (50) ; et

un actionneur télescopique (36) qui permet de déployer la flèche (32) sous le contrôle de l'unité de contrôle (50) ; dans lequel

l'unité de contrôle (50) effectue en outre un procédé de détermination de profil de vitesse radiale pour déterminer un profil de vitesse radiale, le profil de vitesse radiale indiquant la transition d'une vitesse de déplacement de la partie d'extrémité de pointe de la flèche (32) dans une direction radiale de pivotement lorsque le corps pivotant (30) pivote de la position de début de pivotement à la position de fin de pivotement, où dans le profil de vitesse radiale, le rayon de pivotement est augmenté et diminué dans le premier intervalle et le second intervalle,

dans le procédé d'acquisition, l'unité de contrôle (50) acquiert en outre un rayon de pivotement r, le rayon de pivotement r étant une distance horizontale entre le centre de pivotement du corps pivotant (30) et la partie de pointe de la flèche (32) à la position de départ de pivotement,

dans le procédé de détermination de profil de vitesse radiale, l'unité de contrôle (50) détermine le profil de vitesse radiale dans lequel les forces dans la direction radiale de pivotement agissant sur la charge suspendue à une position du rayon de pivotement r à la fin du premier intervalle et à la fin du second intervalle sont équilibrées, et

dans le procédé de contrôle d'actionneur, l'unité de contrôle (50) contrôle l'actionneur de relevage (35) et/ou l'actionneur télescopique (36) afin de permettre à la flèche (32) d'être relevée et/ou déployée de telle sorte que la partie de pointe de la flèche (32) se déplace dans la direction radiale de pivotement à une vitesse indiquée par le profil de vitesse radiale.

5. Dispositif de pivotement selon la revendication 4, dans lequel
dans le procédé de détermination du profil de vitesse radiale, l'unité de contrôle (50) détermine le profil de vitesse radiale dans lequel la charge suspendue se déplace sur le rayon de pivotement r lorsque le corps pivotant (30) pivote de la position de départ de pivotement à la position de fin de pivotement.

6. Dispositif de pivotement selon la revendication 4 ou 5, dans lequel
dans le procédé de détermination du profil de vitesse radiale, l'unité de contrôle (50) détermine une vitesse de déplacement R₀'(t) dans la direction radiale de pivotement de la partie de pointe de la flèche (32) après t seconde à partir du début du pivotement en spécifiant un coefficient r_i (i = 0, ..., 5) de l'équation 2 satisfaisant une condition initiale et une condition terminale du premier intervalle.
[2]

7. Dispositif de pivotement selon la revendication 4 ou 5, dans lequel dans le procédé de détermination du profil de vitesse radiale, l'unité de contrôle (50) détermine une vitesse de déplacement R₀'(t) dans la direction radiale de pivotement de la partie de pointe de la flèche (32) après t seconde à partir du début du pivotement en spécifiant un coefficient b_i (i = 0, ..., 5) de l'équation 3 satisfaisant une condition initiale et une condition terminale du premier intervalle.
[3]

Drawing